Ballistic/Lifting Atmospheric Entry • Straight-line (no gravity) ballistic entry based on altitude, rather than density • Planetary entries (at least a start) • Basic equations of planar motion (with lif) • Lifing equilibrium glide © 2014 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 1 ENAE 791 - Launch and Entry Vehicle Design Terminal Velocity Full form of ODE - d v2 h 2gh s v2 = s d⇤ ⇥ − β sin ⇥ ⇤ At terminal velocity, v = constant v ≡ T h 2gh s v2 = s −β sin ⇥ T ⇤ 2 2gβ sin ⇥ vT = − ⇤ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 2 ENAE 791 - Launch and Entry Vehicle Design “Cannon Ball” γ=-90° Ballistic Entry 6.75” diameter sphere, cD=0.2, VE=6000 m/sec Iron Aluminum Balsa Wood Weight 40 lb 15.6 lb 14.5 oz β (kg/m2) 3938 1532 89 ρmd (kg/m3) 0.555 0.216 0.0125 hmd (m) 5600 12,300 32,500 Vimpact (m/s) 1998 355 0* Vterm (m/sec) 251 156 38 *Artifact of assumption that D mg U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Atmospheric Density with Altitude Pressure=the integral of the atmospheric density in the column above the reference area 1 1 h h h 1 Po = ⇢gdh = ⇢og e− hs dh = ⇢oghs e− s ρ = f(h) o o − o Z Z h i = ⇢ gh [0 1] − o s − Po = ⇢oghs kg Earth: ρ = 1.226 ; h = 7524m; o m3 s Po(calc) = 90, 400 Pa; Po(act) = 101, 300 Pa ρo, Po U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Nondimensional Ballistic Coefficient v h ⇤ ⇤ P ⇢ = exp s o =exp o v 2β sin ⇥ ⇤ 2βg sin γ ⇢ e o ⇥ ✓ o ◆ β βg Let β = (Nondimensional form of ballistic coefficient) ⌘ ⇢ohs Po Note that we are using the estimated value of P = ⇢ gh , b o o s not the actual surface pressure. v 1 ⇤ = exp v ⇤ e 2β sin ⇥ o ⇥ ⇤ h ⇤ 1 β = o s β = crit −sin ⇥ crit −sin ⇥ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 5 ENAE 791 - Launch and Entry Vehicle Design Entry Velocity Trends, γ=-90° Velocity Ratio 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Density Ratio 0.7 0.8 0.9 1 β 0.03 0.1 0.3 1 3 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 6 ENAE 791 - Launch and Entry Vehicle Design Ballistic Entry, Again s = distance along the flight path v, s dv D γ = g sin γ dt − − m D horizontal Again assuming D g, mg dv D 1 = Drag D ρv2Ac dt −m ≡ 2 D dv ρc A = D v2 dt − 2m Separating the variables, dv ⇥ = dt v2 −2β U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Calculating the Entry Velocity Profile dh dh = v sin γ dt = dt ⇒ v sin γ dv ⇤ dv ⇤ = dh = dh v2 −2βv sin ⇥ ⇥ v −2β sin ⇥ dv ⇤o h = e− hs dh v −2β sin ⇥ v h dv ⇤o h = e− hs dh v −2β sin ⇥ ve he h v ⇤ohs h 1 h he ln = e− hs = e− hs e− hs ve 2β sin ⇥ he 2β sin ⇥ − ⇥ ⇥ ⇤ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 8 ENAE 791 - Launch and Entry Vehicle Design Deriving the Entry Velocity Function he ρe Remember that e− hs = 0 ρo ≈ v 1 h = exp e− hs v e 2β sin ⇥ ⇥ We have a parametric entry equation in terms of nondimensional velocity ⇤ ratios, ballistic coefficient, and altitude. To bound the nondimensional altitude variable between 0 and 1, rewrite as v 1 h he = exp e− he hs v e 2β sin ⇥ ⇥ he and β are the⇤ only variables that relate to a specific planet hs U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 9 ENAE 791 - Launch and Entry Vehicle Design Earth Entry, γ=-90° 1 0.9 0.8 0.7 0.6 0.5 0.4 Altitude Ratio 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Velocity Ratio β 0.03 0.1 0.3 1 3 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 10 ENAE 791 - Launch and Entry Vehicle Design Deceleration as a Function of Altitude Start with v 1 h he 1 = exp e− he hs Let B ve ≡ 2β sin ⇥ 2β sin ⇥ ⇥ v h = exp B⇤e− hs ve ⇥ d v h d h = exp Be− hs Be− hs dt ve dt ⇤ ⌅ ⇥ ⇥ dv h B h dh = ve exp Be− hs − e− hs dt hs dt ⇥ ⇥ dh h = v sin γ = v sin γ exp Be− hs dt e U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 11 ENAE 791 - Launch and Entry Vehicle Design Parametric Deceleration dv h B h h = ve exp Be− hs − e− hs ve sin γ exp Be− hs dt hs ⇥ ⇥ ⇥ 2 dv Bve h h = − sin γ e− hs exp 2Be− hs dt hs ⇥ ⇥ 2 dv ve h 1 h = − e− hs exp e− hs dt 2h β β sin ⇥ s ⇥ ⇤ ⌅ 2 ve dv/dt he ⇧ Let nref ⇧, ν , ⇥ ≡ hs ≡ nref ≡ hs 1 ϕ h 1 ϕ h ⇤ = − e− he exp e− he 2β β sin ⇥ ⇥ ⇤ ⌅ U N I V⇧ E R S I T Y O F ⇧ Ballistic and Lifting Atmospheric Entry MARYLAND 12 ENAE 791 - Launch and Entry Vehicle Design Nondimensional Deceleration, γ=-90° 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Altitude Ratio h/he Altitude Ratio 0.2 0.1 0 0 0.05 0.1 0.15 0.2 Deceleratio ν β 0.03 0.1 0.3 1 3 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 13 ENAE 791 - Launch and Entry Vehicle Design Deceleration Equations Nondimensional Form 1 ϕ h 1 ϕ h ⇤ = − e− he exp e− he 2β β sin ⇥ ⇥ ⇤ ⌅ Dimensional Form ⇧ 2 ⇧ ⇤oVe h ⇤ohs h n = − e− hs exp e− hs 2β β sin ⇥ ⇥ ⇤ ⌅ Note that these equations result in values <0 (reflecting deceleration) - graphs are absolute values of deceleration for clarity. U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design Dimensional Deceleration, γ=-90° 120 ) 100 m k ( 80 60 Altitude Altitude 40 20 0 0 500 1000 1500 2000 m Deceleration kg 2 β sec m2 277 922 2767 9224⇥ 27673 ⇥ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 15 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration Returning to shorthand notation for deceleration h h ⇥ = B sin γ e− hs exp 2Be− hs − h ⇥ ⇥ Let η ≡ hs η η ⇥ = B sin γ e− exp 2Be− − d⇤ d⇥ η ⇥ η η d η = B sin γ e− exp 2Be− + e− exp 2Be− d⇥ − d⇥ d⇥ ⇤ ⌅ ⇥ ⇥ ⇥ ⇥ d⇤ η η η η η = B sin γ e− exp 2Be− + e− 2Be− exp 2Be− d⇥ − − − ⇤ ⇥ ⇥ ⇥ ⇥ ⇥⌅ d⇤ η η η = B sin γe− exp 2Be− 1 + 2Be− = 0 d⇥ ⇥ ⇤ ⇥⌅ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 16 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration η η 1 + 2Be− = 0 e = 2B ⇥ − ⇥ η = ln ( 2B) nmax − 1 ⇤nmax = ln − β sin ⇥ ⇥ Converting from parametric to dimensional form gives ⇤ ⇤ h h = h ln − o s nmax s β sin ⇥ ⇥ Altitude of maximum deceleration is independent of entry velocity! U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 17 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration 12 10 8 6 4 2 Altitude of Max Decel h/hs Altitude of 0 0 0.2 0.4 0.6 0.8 1 Ballistic Coefficient β γ -15 -30 -45 -60 -90 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 18 ENAE 791 - Launch and Entry Vehicle Design Magnitude of Maximum Deceleration Start with the equation for acceleration - 1 η 1 η ⇤ = − e− exp e− 2β β sin ⇥ ⇥ and insert the value of η at the point of maximum deceleration ⇤ ⇤ 1 η ⇤ = ln − e− = β sin ⇥ nmax ⇥ − β sin ⇥ ⇥ 1 β sin ⇥ sin γ ⇤nmax = − ⇤ β sin ⇥ exp − ⇥nmax = 2β − ⇤ β sin ⇥ ⌅ ⇒ 2e ⇥ ⇧ 2 ⇧ ve sin γ nmax = ⇧ ⇧ hs 2e Maximum deceleration is not a function of ballistic coefficient! U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 19 ENAE 791 - Launch and Entry Vehicle Design Peak Ballistic Deceleration for Earth Entry 6000 5000 4000 3000 2000 1000 Peak Deceleration (m/sec^2) Peak Deceleration 0 0 5 10 15 Entry Velocity (km/sec) γ -15 -30 -45 -60 -90 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design Velocity at Maximum Deceleration Start with the equation for velocity v 1 η = exp e− v e 2β sin ⇥ ⇥ and insert the value of η at the point of maximum deceleration ⇤ 1 η ⇤ = ln − e− = β sin ⇥ nmax ⇥ − β sin ⇥ ⇥ v β sin ⇥ v ⇤ v = e 0.606v = exp − nmax ∼= e ve 2β sin ⇥ ⇥ ⇥ ⇤e ⇤ Velocity at maximum⇤ deceleration is independent of everything except ve U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 21 ENAE 791 - Launch and Entry Vehicle Design Planetary Entry - Physical Data Radius µ ρo hs vesc (km) (km3/sec2) (kg/m3) (km) (km/sec) Earth 6378 398,604 1.225 7.524 11.18 Mars 3393 42,840 0.0993 27.70 5.025 Venus 6052 325,600 16.02 6.227 10.37 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Comparison of Planetary Atmospheres 100 1 0.01 0 50 100 150 200 250 300 1E-04 1E-06 1E-08 1E-10 1E-12 1E-14 1E-16 Atmospheric Density (kg/m3) Density Atmospheric 1E-18 1E-20 Altitude (km) Earth Mars Venus U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 23 ENAE 791 - Launch and Entry Vehicle Design Planetary Entry Profiles γ = 15o kg − β = 300 km m2 v = 10 e sec V Ve U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 24 ENAE 791 - Launch and Entry Vehicle Design Planetary Entry Deceleration Comparison γ = 15o − km v = 10 e sec kg β = 300 m2 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 25 ENAE 791 - Launch and Entry Vehicle Design Check on Approximation Formulas γ = 15o − km v = 10 e sec kg β = 300 m2 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 26 ENAE 791 - Launch and Entry Vehicle Design Lifting Entry – Free-Body Diagram mv2 v, s L r γ D horizontal mg U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 27 ENAE 791 - Launch and Entry Vehicle Design Dynamics of Lifting Entry Along the velocity vector, dv v2 m = m sin γ mg sin γ D dt r − − Perpendicular to the velocity vector, dγ v2 mv = L + m cos γ mg cos